Are You Ready? Pythagorean Theorem

Transcription

1 SKILL Pythagorean Theorem Teahing Skill Objetive Find the length of the hypotenuse of a right triangle. Have students read the Pythagorean Theorem. Restate the theorem in words, as follows: the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Emphasize that the hypotenuse of a right triangle is ALWAYS the side that is opposite the right angle. Ask: If the lengths of all three sides are found orretly, whih side will always be the longest side? (the hypotenuse) Point out that it does not matter whih leg is represented by a and whih is represented by b, but the hypotenuse must always be represented by. Work the example, stressing that you must square the legs first before you add them. Sine most numbers are not perfet squares, tell students that they may need to simply radials. Work a few examples to remind them of the proess. PRACTICE ON YOUR OWN In exerises, students find the length of the hypotenuse of several right triangles. CHECK Determine that students know how to use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. COMMON ERRORS Students may add the lengths of the legs before squaring them. Students who made more than error in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Alternative Teahing Strategy Objetive Verify the Pythagorean Theorem using a ruler. Materials needed: several piees of lined paper and a ruler Remind students that the Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Tell students they are going to verify the theorem. Have students take one piee of lined paper and fold it arefully in half (vertially), making a distint rease in the paper. Instrut them to unfold the paper. Instrut students to use a ruler to draw a vertial line up the rease inhes long, and a horizontal line at the bottom of the vertial line, inhes long. Next, have students onnet the two lines with a diagonal, forming a right triangle. Using a ruler, students should arefully measure the length of the hypotenuse. Instrut them to label the lengths of the legs, a and b ( and ), and the length of the hypotenuse, (0). Ask: Aording to the Pythagorean Theorem, how are a, b, and related? (a b ). Have students onfirm this by substituting their values into the equation. Repeat the exerise above on separate sheets of paper using the following measurements: ) vertial line inhes; horizontal, inhes (hypotenuse should equal inhes) ) vertial line m; horizontal, m (hypotenuse should equal m) ) vertial line m; horizontal, m (hypotenuse should equal 7 m) When you feel omfortable that students know how to use the Pythagorean Theorem, move on to examples that do not require measurements. Copyright by Holt MDougal. 7 Holt MDougal Geometry

2 Name Date Class SKILL Pythagorean Theorem Pythagorean Theorem If a right triangle has legs of lengths a and b, and a hypotenuse of length, then a b. a hypotenuse legs b Example: Find the length of the hypotenuse of the right triangle. Answer: a b 9 The length of the hypotenuse is. Pratie on Your Own Find the length of the hypotenuse in eah right triangle. If the length is not a whole number, give the answer in simplest radial form Chek Find the length of the hypotenuse in eah right triangle. If the length is not a whole number, give the answer in simplest radial form Copyright by Holt MDougal. 7 Holt MDougal Geometry

3 SKILL Angle Relationships Teahing Skill Objetive Identify angle relationships. Begin by explaining to students that angle relationships often provide information about the measure of the angles. Point out that there are a number of angle postulates and theorems that establish ongruene between ertain types of angles. Emphasize that it is important to be able to identify angle relationships in order to apply those ongruene postulates and theorems. Review the definitions and examples of adjaent angles, vertial angles, omplementary angles, and supplementary angles. Point out the differene between omplementary and supplementary angles. Ask: Whih pair of angles form a straight angle? (supplementary) Instrut students to omplete the pratie exerises. PRACTICE ON YOUR OWN In exerises, students hoose whih desription best fits the angle relationships. In exerises, students use a diagram to give examples of different types of angle relationships. CHECK Determine that students know how to identify angle relationships. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. COMMON ERRORS Students may onfuse the definitions of omplementary and supplementary. Students who made more than errors in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Alternative Teahing Strategy Objetive Identify angle relationships. Materials needed: multiple enlarged opies of the game ards shown below 0 Supplement of 9 0 Complement of 0 9 Angle adjaent to 0 Larger Larger Smaller Angle PQR if P R angle PQS is Q 9 Smaller 0 S Angle 90 Larger Supplement of 00 0 Angle vertial to Larger Equal Tell students they are going to play Larger, Smaller, or Equal. Before you begin, review the definitions and a few examples of adjaent, vertial, omplementary, and supplementary angles. Then, give eah student a set of shuffled game ards. Tell the students that when you say Go, they should math their ards aording to the small numbers in the lower right orner of the ard. Then students should determine whih ard represents the smaller angle and whih represents the larger angle. They should plae the smaller angle in a pile on their left and the larger angle on their right. If the two angles are equal, they should plae them both in a enter pile. The first student to orretly separate their ards wins. Copyright by Holt MDougal. Holt MDougal Geometry

4 Name Date Class SKILL Angle Relationships Adjaent Angles Definition: two angles that share a side and a vertex, but no interior points Example: Angle Relationships Vertial Angles Definition: two angles whose sides are opposite rays Example: and and Complementary Angles Definition: two angles, the sum of whose measures is 90 Supplementary Angles Definition: two angles, the sum of whose measures is 0 Examples: 0 and 0 ; Examples: 0 and 0 ; Pratie on Your Own Cirle the better desription for eah labeled angle pair.. omplementary angles supplementary angles. vertial angles supplementary angles. adjaent angles vertial angles. and omplementary angles supplementary angles Use the diagram to the right to give an example of eah angle pair. F E. adjaent angles. omplementary angles 7. vertial angles. supplementary angles G A C D B Chek Cirle the better desription for eah labeled angle pair. 9. omplementary angles adjaent angles 0. omplementary angles adjaent angles Use the diagram to the right to give an example of eah angle pair.. vertial angles. omplementary angles S T P U. adjaent angles. supplementary angles R Q Copyright by Holt MDougal. Holt MDougal Geometry

5 Name Date Class CHAPTER 0 Complete the rossword puzzle. DOWN. A(n) right triangle has legs of equal length.. The two shortest sides of a right triangle are the.. a b is the Theorem.. The distane around a right triangle is its.. The non-right angles of a right triangle are angles. ACROSS Enrihment The Right Answer. The side of a right triangle that is opposite the right angle is the.. A right triangle has exatly one angle with measure degrees.. The length of the shortest leg of a triangle is the length of the hypotenuse.. A right triangle annot be a(n) triangle.. The length of the hypotenuse of a right triangle with legs of length and is.. The hypotenuse of a right triangle is always the side. 7. The legs of a triangle are.. A right triangle an be a triangle. 7 Copyright by Holt MDougal. 0 Holt MDougal Geometry

6 Answer Key ontinued SKILL 0 ANSWERS: Pratie on Your Own Chek SKILL ANSWERS: Pratie on Your Own Chek SKILL ANSWERS: Pratie on Your Own Chek SKILL ANSWERS: Pratie on Your Own. Yes; ASA. No. Yes; HL. Yes; ASA. Yes; SAS. No Chek 7. No. Yes; HL 9. Yes; SSS SKILL ANSWERS: Pratie on Your Own. A and D. B and C. Yes; orresponding sides are in proportion (:). Yes; orresponding sides are in proportion (7:). Yes; orresponding sides are in proportion (:). No Copyright by Holt MDougal. Holt MDougal Geometry

7 Answer Key ontinued. or. SKILL ANSWERS: Pratie on Your Own. ABC or CBA; right. XYZ or ZYX; obtuse. EDF or FDE; aute. PQR or RQP; obtuse. ABC or CBA; straight. PQR or RQP; aute 7. XYZ or ZYX; straight. EDF or FDE; right 9. ACB or BCA; aute Chek 0. QPR or RPQ; aute. ABC or CBA; straight. SQP or PQS; aute. DEF or FED; right. YXZ or ZXY; aute. ABC or CBA; obtuse SKILL ANSWERS: Pratie on Your Own Chek SKILL ANSWERS: Pratie on Your Own. supplementary angles. vertial angles. adjaent angles. omplementary angles. answers will vary; any two angles that share the vertex (C) and one side. GCF and FCE or ECD and DCB 7. ACG and BCD or GCD and ACB. ACB and BCD or ACB and ACG or ACG and GCD or BCD and DCG or DCE and ECA or DCF and FCA Chek 9. adjaent angles 0. omplementary angles. TPU & QPR or UPQ & TPR. SPR & RPQ. answers will vary; any two angles that share the vertex (P) and a side. UPQ & QPR or UPT & TPR or UPS & SPR or TPU & UPQ or TPS & SPQ or TPR & RPQ SKILL ANSWERS: Pratie on Your Own. h. a,. h. e. h Copyright by Holt MDougal. Holt MDougal Geometry